\newproblem{lay:4_5_32}{
  % Problem identification
	\begin{large}
	  \hspace{\fill}\newline
    \textbf{Lay, 4.5.32}
	\end{large}
	\\
  \ifthenelse{\boolean{identifyAuthor}}{\textit{Carlos Oscar Sorzano, Aug. 31st, 2013} \\}{}

  % Problem statement
	Consider finite-dimensionals spaces $V$ and $W$, and a linear transformation $T:V\rightarrow W$. Let $H$ be a nonzero subspace of $V$, and let $T(H)$ be the set
	of images of vectors in $H$. Suppose, additionally, that $T$ is a one-to-one mapping. Prove that $\dim\{T(H)\}=\dim\{H\}$. If $T$ happens to be a one-to-one
	mapping of $V$ onto $W$, then $\dim\{V\}=\dim\{W\}$. Isomorphic finite-dimensional vector spaces have the same dimension.
}{
  % Solution
	If $T$ is a one-to-one mapping, it means that it maps linearly independent sets from $V$ to $W$ and viceversa. If the dimension of $H$ is $k$, then any basis of $H\subseteq V$
	has $k$ linearly independent vectors, that are mapped by $T$ onto $k$ linearly independent vectors of $W$. So they are also a basis of $T(H)$, and consequently $\dim\{T(H)\}=k$.
}
\useproblem{lay:4_5_32}
\ifthenelse{\boolean{eachProblemInOnePage}}{\newpage}{}
